This paper presents an efficient hardware algorithm for variable-precision logarithm. The algorithm uses an iterative technique that employs table lookups and polynomial approximations. Compared to similar algorithms, it reduces the number of fixed-precision operations by avoiding full precision computations and dynamically varying the precision of intermediate results. It also uses significantly smaller tables than related algorithms. Fora specified hardware implementation, the algorithm requires fewer than 2 x L**2 fixed-precision multiplications to evaluate the logarithm to L words of precision. An error analysis for the algorithm is also presented.